In the presence of vacuum, the physical entropy for polytropic gases behave singularly, and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the one-dimensional heat conductive compressible Navier-Stokes equations, the global well-posedness of strong solutions and uniform boundedness of the corresponding entropy are established, as long as the initial density vanishes only at far fields with a rate no more than . The main tools of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier-Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi–type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy. © 2021 Wiley Periodicals LLC.

中文翻译：

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远场真空一维导热可压缩纳维-斯托克斯方程的熵有界解

在真空存在下，多变气体的物理熵表现异常，因此研究其动力学是一个挑战。本文表明，熵的有界性可以传播到任何有限时间，前提是初始真空仅存在于远场，初始密度衰减足够慢。更准确地说，对于一维导热可压缩 Navier-Stokes 方程的 Cauchy 问题，只要初始密度仅在远场消失率不超过. 证明熵均匀有界的主要工具是一些为导热可压缩 Navier-Stokes 方程精心设计的奇异加权能量估计，以及用于某些类别的退化抛物方程的精细 De Giorgi 类型迭代技术。De Giorgi 型迭代在建立熵的下界和上界时对不同的方程进行。© 2021 威利期刊有限责任公司。